Of course it does.
If you rolled 1000000 d6s, the odds of getting a 6 aren't 1 in 6. That would imply that 5 times out of six you would get no sixes at all, which is clearly silly.
ah, now we enter the subject of math, or that which I am not an expert.
If you were going to pay me $20 if a six showed up on any of the dice that I might roll, and then allowed me to roll 1 or more dice at the same time, I would of course choose to roll more dice, so as to improve my chances.
Thus thinking that, somebody once pointed out that I was wrong, and that regardless of how many dice I rolled, the actual chance of a 6 showing up on any of them remained 1 in 6. I thought that was ridiculous, until I ran all the iterations and confirmed it (remember, not a math guy).
I would certainly accept that my primitive experiment was flawed (I wrote a bit of code to loop and count the occurances). But it seemed to confirm what the somebody said and demonstrated through a bit of math.
so here is your chance to mathematically resolve the matter. I'm interested in seeing it done correctly.
(and yes, the thread has forked, but I think the mystery of the dice does relate to life on other planets as a matter of analogous statistics)